Quantum quench in p+ip superfluids: Winding numbers and topological - - PowerPoint PPT Presentation

Matthew S. Foster,1 Maxim Dzero,2 Victor Gurarie,3 and Emil A. Yuzbashyan4

1 Rice University, 2 Kent State University,

3 University of Colorado at Boulder, 4 Rutgers University April 26th, 2013

Quantum quench in p+ip superfluids: Winding numbers and topological states far from equilibrium

What is a “quantum quench”? A non-adiabatic perturbation to a closed quantum many-particle system

Quantum Quench: Coherent many-body evolution Quantum quench protocol 1. Prepare initial state e.g.

• : ground state of
• Assume excitation gap

Quantum quench protocol 1. Prepare initial state 2. “Quench” the Hamiltonian: Non-adiabatic perturbation Quantum Quench: Coherent many-body evolution

Quantum quench protocol 1. Prepare initial state 2. “Quench” the Hamiltonian: Non-adiabatic perturbation 3. Exotic excited state, coherent evolution Quantum Quench: Coherent many-body evolution

Quantum Quench: Coherent many-body evolution  Decoherence, dissipation…

Quantum Quench: Coherent many-body evolution

Bloch, Dalibard, Zwerger 2008

Experimental Example:

Collapse and revival of matter wave interference: SF to Mott quench in a boson atom optical lattice

Greiner, Mandel, Hänsch, and Bloch 2002

New fields of non-equilibrium dynamics 1. Nanostructures, qubits 2. Ultrafast spectroscopy 3. Ultracold atoms

• Extreme isolation → Long relaxation times
• Highly tunable

Experimental Example:

Quantum Newton’s Cradle for trapped 1D Bose Gas

Quantum Quench: Coherent many-body evolution

Kinoshita, Wenger, and Weiss 2006

• Thermalization
• Quantum critical scaling
• Post-quench quasiparticle distribution functions
• Post-quench evolution of correlation functions
• Transverse field Ising

Cincio, Dziarmaga, Rams, Zurek 2007; Rossini, Silva, Mussardo, Santoro 2009 Calabrese, Essler, Fagotti 2011; Schuricht and Essler 2012; Calabrese, Essler, and Fagotti 2012

• Lieb-Liniger

Demler, Imambekov, Kormos, Iyer, Andrei, Gritsev, Mossel, Caux, Buljan, Pezer, Gasenzer, Konik...

• “Quantum” solitons

Foster, Altshuler, and Yuzbashyan (2010), Foster, Berkelbach, Reichman, and Yuzbashyan (2011); Neuenhahn, Polkovnikov, Marquardt (2012)

The theoretical story so far: Quantum quenches

Rigol, Dunjko, Yurovsky, Olshannii (2007) … Calabrese and Cardy (2006); Gritsev, Demler, Lukin, Polkovnikov (2007) ... Rigol, Dunjko, Olshannii (2008); Rossini, Silva, Mussardo, Santoro (2009)... Polkovnikov (2005), De Grandi, Gritsev, Polkovnikov (2010) …

Spinless (or spin-polarized) fermions in 2D: P-wave BCS Hamiltonian

P-wave superconductivity in 2D

Spinless (or spin-polarized) fermions in 2D: P-wave BCS Hamiltonian Anderson pseudospins

P-wave superconductivity in 2D

Spinless (or spin-polarized) fermions in 2D: P-wave BCS Hamiltonian Anderson pseudospins

Time-reversal particle pair No pair

P-wave superconductivity in 2D

{k,-k} vacant {k,-k} vacant

Spinless (or spin-polarized) fermions in 2D: P-wave BCS Hamiltonian Anderson pseudospins Fermi liquid state: Discontinuous domain wall

P-wave superconductivity in 2D

Spinless (or spin-polarized) fermions in 2D: P-wave BCS Hamiltonian Anderson pseudospins Superconducting state: Smooth domain wall

P-wave superconductivity in 2D

• P. Anderson 1958

Spinless (or spin-polarized) fermions in 2D: P-wave BCS Hamiltonian “P + i p” superconducting state: Skyrmion pseudospin texture

P-wave superconductivity in 2D

• G. E. Volovik

1988; Read and Green 2000

Spinless (or spin-polarized) fermions in 2D: P-wave BCS Hamiltonian “P + i p” superconducting state:

Fully gapped, non s-wave

P-wave superconductivity in 2D

solidforms.com

Spinless (or spin-polarized) fermions in 2D: P-wave BCS Hamiltonian “P + i p” superconducting state:

P-wave superconductivity in 2D

At fixed density n:

• µ is a monotonically

decreasing function of ∆0 BCS BEC

Pseudospin winding number Q :

Topological superconductivity in 2D

BCS BEC

2D Topological superconductor

• Fully gapped when µ ≠ 0
• Weak-pairing BCS state

topologically non-trivial

• Strong-pairing BEC state

topologically trivial

• G. E. Volovik 1988; Read and Green 2000

Pseudospin winding number Q :

Topological superconductivity in 2D

• G. E. Volovik 1988; Read and Green 2000

Retarded GF winding number W (i.e., compute G in BdG MFT):

• W = Q in ground state
• W ≠ 0 signals presence of chiral edge states

Niu, Thouless, and Wu 1985

• G. E. Volovik 1988

Topological superconductivity in 2D

Topological signatures: Majorana fermions

• 1. Chiral 1D Majorana edge states

quantized thermal Hall conductance

• 2. Isolated Majorana zero modes

in type II vortices

• J. Moore

Realizations?

• 3He-A thin films, Sr2RuO4(?)
• 5/2 FQHE: Composite fermion Pfaffian
• Cold atoms
• Polar molecules
• S-wave proximity-induced SC on surface of 3D Z2 Top. Insulator

Moore and Read 1991, Read and Green 2000 Fu and Kane 2008 Gurarie, Radzihovsky, Andreev 2005; Gurarie and Radzihovsky 2007 Zhang, Tewari, Lutchyn, Das Sarma 2008; Sato, Takahashi, Fujimoto 2009; Y. Nisida 2009 Cooper and Shlyapnikov 2009; Levinsen, Cooper, and Shlyapnikov 2011 Volovik 1988, Rice and Sigrist 1995

2D weak-pairing BCS p+ip superconductor: Fully-gapped, “strong” topological state (class D) Robust against

• Weak (enough) disorder
• Weak interaction perturbations (e.g., pair-breaking terms)

Stability of topological order against

• 1. Strong disorder?
• 2. Hard non-equilibrium driving?

Topological protection

2D weak-pairing BCS p+ip superconductor: Fully-gapped, “strong” topological state (class D) Robust against

• Weak (enough) disorder
• Weak interaction perturbations (e.g., pair-breaking terms)

Stability of topological order against

• 1. Strong disorder?
• 2. Hard non-equilibrium driving?

Topological Order vs. Quantum Quench (Fight!)

Topological protection …?

• Entanglement entropy survival in the toric code following quench

Rahmani and Chamon 2010

• Kibble-Zurek excitation probability in adiabatic p+ip chain quench

DeGottardi, Sen, and Vishveshwara 2011

2D P-wave BCS Hamiltonian…“hard”

P-wave superconductivity in 2D: Dynamics

Ibañez, Links, Sierra, and Zhao (2009): Chiral 2D P-wave BCS Hamiltonian

• Same p+ip ground state, non-trivial (trivial) BCS (BEC) phase
• Integrable (hyperbolic Richardson model)
• Reduces to 1D:

Claim: for p+ip initial state, self-consistent mean-field dynamics are identical to “real” p-wave Hamiltonian

P-wave superconductivity in 2D: Dynamics

Richardson (2002) Dunning, Ibanez, Links, Sierra, and Zhao (2010) Rombouts, Dukelsky, and Ortiz (2010)

BCS BEC

P-wave Quantum Quench

• Initial p+ip BCS or BEC state:
• Post-quench Hamiltonian:
• Quench parameter:

P-wave Quantum Quench

BCS BEC

• Initial p+ip BCS or BEC state:
• Post-quench Hamiltonian:
• Quench parameter:

Heisenberg spin equations of motion: Self-consistent mean field theory (thermodynamic limit) Identical to self-consistent (time-dependent) Bogoliubov-de Gennes

Chiral P-wave BCS: Dynamics

Classical spin equations of motion:

Chiral P-wave BCS: Dynamics

Classical spin equations of motion: Ground state: spins aligned with field!

Chiral P-wave BCS: Dynamics

Classical spin equations of motion: Ground state:

• 1. “Gap equation”
• 2. Chemical potential vs density

(Absorb linear divergence into G at QCP)

Chiral P-wave BCS: Dynamics

Lax vector components, “norm”

Chiral P-wave BCS: Lax construction

Foster, Dzero, Gurarie, Yuzbashyan (unpublished)

Lax vector components, “norm” Generalized Gaudin algebra

Chiral P-wave BCS: Lax construction

• M. Gaudin 1972, 1976, 1983

Foster, Dzero, Gurarie, Yuzbashyan (unpublished)

Lax vector components, “norm” Lax vector norm: Generates integrals of motion From Gaudin algebra: BCS Hamiltonian:

Chiral P-wave BCS: Lax construction

Lax vector components, “norm” Conserved spectral polynomial: Key to understanding ground state and quench dynamics

Chiral P-wave BCS: Lax construction

Yuzbashyan, Altshuler, Kuznetsov, and Enolskii (2005)

In the ground state (zero quench):

Ground state: Lax roots

In the ground state (zero quench):

Ground state: Lax roots

In the ground state (zero quench): One isolated pair of roots ; (N – 1) positive, real, doubly-degenerate roots Gap, chemical potential encoded in isolated roots

Ground state: Lax roots

P-wave quantum quench: Lax roots

Quench

• Initial p+ip BCS or BEC state:
• Post-quench Hamiltonian:
• Quench parameter:

Roots: Strong-to-weak quench #1 (β > 0)

P-wave quantum quench: Lax roots

Quench

100 spins (numerics)

1 isolated pair

• Initial p+ip BCS or BEC state:
• Post-quench Hamiltonian:
• Quench parameter:

Roots: Strong-to-weak quench #1 (β > 0)

Quench

100 spins (numerics)

No isolated pair

P-wave quantum quench: Lax roots

• Initial p+ip BCS or BEC state:
• Post-quench Hamiltonian:
• Quench parameter:

Roots: Strong-to-weak quench #2 (β > 0)

Quench

100 spins (numerics)

2 isolated pairs

P-wave quantum quench: Lax roots

• Initial p+ip BCS or BEC state:
• Post-quench Hamiltonian:
• Quench parameter:

Roots: Weak-to-strong quench #1 (β < 0)

Quench dynamics: Isolated roots determine phase diagram

Via continuum limit of the spectral polynomial, only possibilities: (same as s-wave) 1) No isolated pairs. Gap decays to zero. 2) One isolated pair. Effective one-spin spectral polynomial: Gap goes to a constant ; non-eqlm chemical potential 3) Two isolated pairs. Effective two-spin spectral polynomial: Gap oscillates according to elliptic EOM (2-spin problem).

Yuzbashyan, Altshuler, Kuznetsov, and Enolskii 2005 Yuzbashyan, Tsyplyatyev, and Altshuler 2005 Barankov and Levitov 2006 Dzero and Yuzbashyan 2006 Foster, Dzero, Gurarie, Yuzbashyan (unpublished)

Spectral polynomial for a quench: where

Isolated roots: Thermodynamic limit

Roots: where

Isolated roots: Thermodynamic limit

Exact quench phase diagram: Strong to weak, weak to strong quenches

Region I:

Zero isolated roots. Gap decays to zero.

Region II:

One isolated pair of roots. Gap goes to a constant.

Region III:

Two pairs of isolated roots. Gap oscillates.

Foster, Dzero, Gurarie, Yuzbashyan (unpublished)

Gap dynamics for reduced 2-spin problem: Parameters completely determined by two isolated root pairs Initial parameters:

Region III weak to strong quench dynamics: Oscillating gap

*

Blue curve: classical spin dynamics (numerics 5024 spins) Red curve: solution to Eq. ( )

*

Foster, Dzero, Gurarie, Yuzbashyan (unpublished)

Gap dynamics for reduced 2-spin problem: Parameters completely determined by two isolated root pairs

Region III weak to strong quench dynamics: Oscillating gap

*

Initial parameters:

Blue curve: classical spin dynamics (numerics 5024 spins) Red curve: solution to Eq. ( )

*

Foster, Dzero, Gurarie, Yuzbashyan (unpublished)

Pseudospin winding number Q: Dynamics

Pseudospin winding number Q Chiral p-wave model: Spins along arcs evolve collectively:

Pseudospin winding number Q: Dynamics

Pseudospin winding number Q

∴ Winding number is again given by

Well-defined, so long as spin distribution remains smooth (no Fermi steps)

Pseudospin winding number Q: Unchanged by quench!

Foster, Dzero, Gurarie, Yuzbashyan (unpublished)

Pseudospin winding number Q: Unchanged by quench!

“Topological” Gapless State

Foster, Dzero, Gurarie, Yuzbashyan (unpublished)

Gapless Region A,

Post-quench spin distribution: Gapless phase

Foster, Dzero, Gurarie, Yuzbashyan (unpublished)

Q = 0

Gapless Region B,

Q = 1

As t → ∞, spins precess around “effective ground state field” determined by the isolated roots. Gapless phase : Compute γ(ε) from the conservation of the Lax norm

“Topological” gapless phase

Decay of gap (dephasing): 1) Initial state not at the QCP (|∆| ≠ ∆QCP, µ ≠ 0): 2) Initial state at the QCP (|∆| = ∆QCP, µ = 0): Gapless Region A,

Q = 0

Gapless Region B,

Q = 1

Foster, Dzero, Gurarie, Yuzbashyan (unpublished)

Gapless Region A,

Q = 0

Gapless Region B,

Q = 1

Foster, Dzero, Gurarie, Yuzbashyan (unpublished)

Retarded GF winding number W :

• Same as pseudospin

winding Q in ground state

• Signals presence of chiral

edge states in equilibrium

New to p-wave quenches:

• Chemical potential µ(t)

also a dynamic variable!

• Phase II:

Niu, Thouless, and Wu 1985

• G. E. Volovik 1988

Retarded GF winding number W: Dynamics

Retarded GF winding number W: Dynamics

Purple line:

Quench extension of topological transition

“winding/BCS” “non-winding/BEC”

Foster, Dzero, Gurarie, Yuzbashyan (unpublished)

Retarded GF winding number W :

• Same as pseudospin

winding Q in ground state

• Signals presence of chiral

edge states in equilibrium

Niu, Thouless, and Wu 1985

• G. E. Volovik 1988

Foster, Dzero, Gurarie, Yuzbashyan (unpublished)

Retarded GF winding number W: Dynamics

 W ≠ Q out of equilibrium!  Winding number W can change following quench across QCP  Result is nevertheless quantized as t → ∞  Edge states can appear or disappear in mean field Hamiltonian spectrum (Floquet)

 W ≠ Q out of equilibrium!  Winding number W can change following quench across QCP  Result is nevertheless quantized as t → ∞  Edge states can appear or disappear in mean field Hamiltonian spectrum (Floquet)  …Does NOT tell us about

• ccupation of edge or bulk

states

Foster, Dzero, Gurarie, Yuzbashyan (unpublished)

Retarded GF winding number W: Dynamics

Pseudospin winding Q Ret GF winding W

Bulk signature? “Cooper pair” distribution

As t → ∞, spins precess around “effective ground state field” determined by the isolated roots. Gapped phase: Parity of distribution zeroes odd when Q (pseudospin) ≠ W (Ret GF)

Post-quench “Cooper pair” distribution: Gapped phase II

Gapped Region C,

Q = 0 W = 1

Gapped Region D,

Q = 1 W = 1

• Cooper pair distribution γ(ε) in principle measurable in

RF spectroscopy

• Parity of zeroes: new out-of-eqlm winding number,

non-trivial when Q ≠ W

Post-quench “Cooper pair” distribution: Gapped phase II

Summary and open questions

• Quantum quench in p-wave superconductor investigated
• Dynamics in thermodynamic limit exactly solved via classical integrability
• Quench phase diagram, exact asymptotic gap dynamics

1) Gap goes to zero (pair fluctuations) 2) Gap goes to non-zero constant 3) Gap oscillates same as s-wave case

• Pseudospin winding number Q is unchanged by the quench, leading to

“gapless topological state”

• Retarded GF winding number W can change under quench; asymptotic

value is quantized. Corresponding HBdG possesses/lacks edge state modes (Floquet)

• Parity of zeroes in Cooper pair distribution is odd whenever Q ≠ W